Bayesian Single Sampling Acceptance Plans for Finite Lot Sizes

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Abstract
Summary: We discuss a general model for acceptance/rejection decisions regarding finite populations. The model is based on an exchangeable prior distribution of the population characteristics and additive costs. A fundamental property of the model is that the original two-decision problem regarding the characteristics of the finite population is equivalent to a classical decision problem regarding a parameter in the prior distribution of population characteristics. In the general discussion of the model we verify some simple results for the optimal sampling scheme, viz. that the expected decision cost per individual is decreasing and the optimal sample size is increasing with the population size. A substantial part of the paper is devoted to the study of monotonely regular problems, i.e. problems admitting a sufficient statistic with a monotone likelihood-ratio. A main result of the discussion is that the optimal decision criterion is a monotone function of the sample size. It is shown that the theory describes the standard situations in quality control with lot-by-lot sampling, where the distribution of the item characteristics belongs to an exponential family, and in an example we suggest a simple algorithm for determination of the Bayesian sampling scheme.